If z is a complex number such that$\left|z\right|=1,$ then the maximum value of $|z^4+z^3-2z^{2}i+z+1|$ is.

Let  $u=\left|z^2\right||z^2+\frac{1}{z^2}+z+\frac{1}{z}-2i|$
 $=|(z^2+{\overline{z}}^2)+(z+\overline{z})-2i|$
 $=|{(z+\overline{z})}^2-2z\overline{z}+(z+\overline{z})-2i|$
let $z=x+iy$
 $\therefore u=|{\left(2x\right)}^2-2+2x-2i|$
 $\begin{array}{l}u=2|2x^2+x-1-i|\\ u^2=4({(2x^2+x-1)}^2+1)\end{array}$
 $\begin{array}{l}\because \left|z\right|=1\therefore x^2+y^2=1\\ \therefore -1\le x\le 1\end{array}$
Now  $t=2x^2+x-1=2(x^2+\frac{1}{2}x-\frac{1}{2})$
  $=2({(x+\frac{1}{4})}^2-\frac{9}{16})$
 
 $\frac{-9}{8}\le t\le 2\therefore u_{\max }^2=20$
$u_{\max }=\sqrt{20}$